As part of our ongoing NTT SoFI survey for variability in young free-floating planets and low-mass brown dwarfs, we detect significant variability in the young, free-floating planetary-mass object PSO J318.5–22, likely due to rotational modulation of inhomogeneous cloud cover. A member of the 23 ± 3 Myr β Pic moving group, PSO J318.5–22 has Teff = {1160}-40+30 K and a mass estimate of 8.3 ± 0.5 MJup for a 23 ± 3 Myr age. PSO J318.5–22 is intermediate in mass between 51 Eri b and β Pic b, the two known exoplanet companions in the β Pic moving group. With variability amplitudes from 7% to 10% in JS at two separate epochs over 3–5 hr observations, we constrain the rotational period of this object to >5 hr. In KS, we marginally detect a variability trend of up to 3% over a 3 hr observation. This is the first detection of weather on an extrasolar planetary-mass object. Among L dwarfs surveyed at high photometric precision (<3%), this is the highest amplitude variability detection. Given the low surface gravity of this object, the high amplitude preliminarily suggests that such objects may be more variable than their high-mass counterparts, although observations of a larger sample are necessary to confirm this. Measuring similar variability for directly imaged planetary companions is possible with instruments such as SPHERE and GPI and will provide important constraints on formation. Measuring variability at multiple wavelengths can help constrain cloud structure. Based on observations made with ESO Telescopes at the La Silla Paranal Observatory under program ID 095.C-0590.